Compound of five octahedra


Compound of five octahedra
Compound of five octahedra
Compound of five octahedra.png
Type Regular compound
Index UC17, W23
Elements
(As a compound)
5 octahedra:
F = 40, E = 60, V = 30
Dual compound Compound of five cubes
Symmetry group icosahedral (Ih)
Subgroup restricting to one constituent pyritohedral (Th)
Stellation diagram Stellation core Convex hull
Stellation facets Icosahedron.png
Icosahedron
Icosidodecahedron.png
Icosidodecahedron

This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876.

Contents

As a stellation

It is the second stellation of the icosahedron, and given as Wenninger model index 23.

It can be constructed by a rhombic triacontahedron with rhombic-based pyramids added to all the faces, as shown by the five colored model image.

As a compound

It can also be seen as a polyhedral compound of five octahedra arranged in icosahedral symmetry (Ih).

It shares its edges and half of its triangular faces with the compound of five tetrahemihexahedra.

Compound of five octahedra.png
Compound of five octahedra
UC18-5 tetrahemihexahedron.png
Compound of five tetrahemihexahedra

As a facetting

It is a facetting of the icosidodecahedron, as shown below:

Small icosiicosahedron-in-icosidodecahedron.gif

At left only one octahedron is shown; all five are shown at right.

References

  • Peter R. Cromwell, Polyhedra, Cambridge, 1997.
  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. 
  • Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR676126  (1st Edn University of Toronto (1938))
  • E. Hess 1876 Zugleich Gleicheckigen und Gleichflächigen Polyeder, Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg 11 (1876) pp 5–97.

External links

Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
Icosahedron Small triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.